International Journal of Mechanical Sciences 49 (2007) 230–238 Heating of a uniform wafer disk K.A. Seffen à , R.A. McMahon Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK Received 31 July 2006; accepted 10 August 2006 Available online 26 September 2006 Abstract The closed-form heating response of a thin uniform circular wafer is obtained, in view of a new processing method for semiconductor materials. A strain energy formulation is obtained expeditiously using Gaussian curvature and associated structural concepts. The method is developed for generally curved wafers, which accounts for flat, spherical, cylindrical and twisted shapes. Solutions for the first two types become available in closed form, and the deformation can exhibit a sudden change in axi-symmetrical response or a snap- through buckling, or both: for the latter two types, a numerical solution points to progressive deformation in both without buckling. All results are shown to compare rather well with finite element analysis. r 2006 Elsevier Ltd. All rights reserved. Keywords: Heating; Semi-conductor wafer; Gaussian curvature; Buckling; Finite elements 1. Introduction Recent innovations in the processing of semiconductor materials pose, rather fortuitously, some interesting and novel challenges in structural mechanics, which are addressed in this study. Of interest here is the rapid treatment of a film (E35 nm) of cubic silicon carbide (SiC) on a thin disk (100 mm  0.5 mm) of silicon by flashlamp irradiation [1] for ultimate use in high-integrity electrical devices. After being uniformly heated to 1000 1C, the composite silicon-SiC wafer suffers a pulse of high-energy radiation for about 20 ms: the intensity and shortness of pulse melts the silicon precisely beneath the film, enabling the dissipation of locked-in stresses, the removal of defects in the original SiC, and the subsequent epitaxial growth of superior SiC. Unfortunately, severe thermal gradients distort the wafer transversely, possibly seeding irrevocable shape changes and damage. However, the promise of drastically reduced processing times motivates an under- standing of the limitations of manufacturing by this method, in pursuit of refinement and, ultimately lower production costs. As a start, the non-linear thermal response for a given set of pulse attributes can be obtained by a diffusion equation approach [2] but deflexions do not feature, and accounting for them in a fully coupled, thermal elastic framework is not trivial. A simpler route assumes a de-coupled and quasi-statical performance, and is reasonable in view of the relative thinness of the wafer. The extremely thin SiC and melt layers do not disrupt the material homogeneity and isotropy within the wafer, which has the properties of silicon given in Table 1. Determining the wafer response under spatial variations of temperature is now sufficient, and here, they persist only in the direction z measured normal to the middle surface of the wafer, and are denoted by T(z). In the absence of constraints, the wafer deflects initially into a uniform spherical ‘‘cap’’, of a curvature, k, equal to [3]. k T ¼ Z t=2 t=2 zaT ðzÞ dz , Z t=2 t=2 z 2 dz, (1) where a is the linear coefficient of thermal expansion, and t is the total thickness. For moderately large deflexions under increasing k T , geometrical compatibility conditions demand the build-up of in-plane strains, and k departs from Eq. (1) as the wafer begins to stiffen. Despite the simple geometry, a solution within an exact large-deflexion plate formulation [4] becomes intractable for constant ARTICLE IN PRESS www.elsevier.com/locate/ijmecsci 0020-7403/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2006.08.003 à Corresponding author. Tel.: +44 1223 764137; fax: +44 1223 332662. E-mail address: kas14@cam.ac.uk (K.A. Seffen).